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Full text of "Treatise On Analysis Vol-Ii"

26       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

The lower envelope of an infinite family of lower semicontinuous functions
is not necessarily lower semicontinuous. As an example, for each rational
number r let/r denote the characteristic function of R  {r}, which is lower
semicontinuous (12.7.4). The lower envelope / of the denumerable family
C/r)reQ takes the value 0 at each rational number and the value 1 at each
irrational number (in other words, it is the " Dirichlet function" 1  <pQ, cf.
(3.11.6)), and is not lower semicontinuous at irrational points.

The following proposition is a partial converse of (12.7.7):

(12.7.8) Let E be a separable, metrizable, locally compact space, and let f be
a lower semicontinuous mapping ofE into R. Suppose that there exists a finite
real valued function g, continuous on E and with compact support, such that
f^.g (which implies that/(.x) ^ 0 in the complement of some compact set).
Then there exists an increasing sequence (/,) of'finitereal valued functions, con-
tinuous on E and with compact supports, such that f = sup fn.

n

Replacing/if necessary by/ g (which is everywhere defined), we may
assume that/g 0. It is then enough to show that there exists a denumerable
family of finite real-valued continuous functions with compact supports, with
/as upper envelope. For it we arrange this family in a sequence (gn) and take
/ = sup(#1? ..., gn), it is clear that the/, satisfy the required conditions.

Let (Un) be an increasing sequence of relatively compact open sets in E,
such that E = (J Un (3.18.3). If gn is the function which agrees with/on UM

71

and is zero on E  Un, then clearly/= sup#n, and gn is lower semicontin-

n

uous. Hence if gn is the upper envelope of a denumerable set Dn of finite real-
valued continuous functions with compact supports, then / will be the upper
envelope of (J DM (2.3.6). Hence we may restrict ourselves to the case where

n

f has compact support.

Suppose first of all that/= <pA, where A is a relatively compact open
set. Put/M(;c) = n - inf(d(x, E - A), l/ri). This is a finite, continuous, positive-
valued function on E, with compact support (3.11.8). Moreover, we have
/n(X) =/W whenever x e E  A or d(x, E  A) ^ l/n. It follows immediately
that/=sup/n (3.8.9).

n

In the case where/^ 0 and has compact support, we can restrict ourselves
to the case where 0 g/^ 1, provided that the functions/, are chosen so that
0 ^fn(x) < 1 for all x e E, For in the general case the function h =//(! +/)
(taken to have the value 1 at points where/(X) = + oo) is lower semicontinu-
ous (cf. (3.3.2)). If we have h = sup hn, with hn continuous and hn(x) < I forother words, Supp(/) is the